Plasma and Fusion Research: Rapid Communications

Volume 1, 015 (2006)

A Shear-Flow Instability Relevant to Advanced
Tokamak Operation
Tomoya TATSUNO and William DORLAND
Center for Scientific Computation and Mathematical Modeling,
The University of Maryland, College Park, Maryland 20742-3289, USA(Received 1 February 2006 / Accepted 17 February 2006)
An ideal magnetohydrodynamic instability which is driven by sheared flow and which may be relevant toreversed-shear advanced tokamak operation is described. If there is flow shear at the qmin surface, relatively weakvelocity shear can drive this instability, with a time scale of the flow. The flows may be significantly slower than
the ambient poloidal Alfvén velocity, and no inflection point is needed. Thus the time scale of the instability
may be significantly longer than that of the poloidal Alfvén transit time, and it might account for disruption ofreversed-shear discharges recently observed in JT-60U.
c 2006 The Japan Society of Plasma Science and Nuclear Fusion Research

Keywords: MHD, shear flow, instability, reversed-shear tokamak, disruption
DOI: 10.1585/pfr.1.015
In recent advanced tokamak experiments, disruption
is observed which is specific for reversed-shear configurations [1]. When qmin value approaches to a certain integral
value, the plasma becomes unstable, and experiences disruption, which shows a longer than ideal but shorter thanresistive time scale in some cases. In this paper, we present
single helicity, nonlinear simulations of an ideal, low mode
number, magnetohydrodynamic (MHD) instability that isdriven by the combined eﬀect of sheared flow and nonmonotonic field profile. More details will be presented in alonger, separate paper. This instability is a good candidate
for explaining the experimental observations. The growthrate of this instability is governed by the magnitude of theflow shear. It is larger than that of the typical resistive instability and smaller than that of the typical magnetic-field
driven ideal instability. It is interesting to note that the flow
profile we use here is itself stable without magnetic field,
and any magnetic field profile is itself stable in slab geometry without background flow, so the instability is truly acombined eﬀect of shear flow and magnetic field.
The instability we discuss here was first pointed out by
Stern [2], who realized that an inhomogeneous magnetic
field may break the conservation of vorticity, just as the inflection point does in the case of Kelvin-Helmholtz (KH)
instability [3]. Kent [4] extended this idea to the caseof symmetric flow profile, and obtained several instability
conditions in terms of the background values at the edge.
Later, Chen and Morrison [5] extended Kent’s work. We
review the understanding of this instability, present the first
nonlinear simulations (in slab geometry), and point out that
this instability may actually be relevant to reversed-shear
discharge of the advanced tokamak.

Since the instability is global and depends strongly on
the detail of the field profile, it is diﬃcult to make a quantitative discussion by analytic calculation. We cannot discuss the instability in the infinite domain, since the infinite
diﬀerence of the shear flow at both edges formally prevents exponential instability due to its stretching (shearing)eﬀect [6]. On the other hand, a periodic domain wouldinevitably introduce inflection point (so as the case of finite flow diﬀerence in the infinite domain), which brings us
confusion to distinguish the phenomena from KH instability [7]. In practice, the overall growth rate has never beenobtained by analytic calculation even for a particular profile in the simplest slab geometry. Here we present the firstlinear and nonlinear results from numerical simulations inthe slab geometry.
In order to simulate the incompressible motion of theplasma, we developed a pseudospectral code which solves
two-dimensional reduced MHD equations with vorticitystream function and flux function formulation. To resolve
the nonperiodic dynamics in the radial direction with spectral techniques, we use a Chebyshev polynomial basis. Our
simulations show that nonlinear evolution of the reversedshear tokamak example may lead to disruption with a timescale slower than Alfvén time and faster than the resistive
one, which might account for the disruption recently observed in JT-60U [1].
We discuss the incompressible motion of shear flow
plasmas in slab geometry. Taking the variation of background field in x (radial) direction, we solve Strauss’ reduced MHD equations with finite viscosity and resistivity
in the single-helicity (two-dimensional) limit [8],∂t ∇⊥2 U + u · ∇∇⊥2 U = B · ∇∇⊥2 A + ν∇⊥2 ∇⊥2 U,(1)author’s e-mail: tatsuno@umd.edu

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c 2006 The Japan Society of Plasma
Science and Nuclear Fusion ResearchPlasma and Fusion Research: Rapid Communications

Volume 1, 015 (2006)

Fig. 1 Time evolutions of ||U1 ||, and enstrophy, total (E T ) andmagnetic (EM ) energy normalized by their initial totalvalues in the simulation with ν = η = 6 × 10−5 and768 × 768 grid points. Also shown is the evolution of||U1 || with ν = η = 1.2 × 10−4 and 512 × 512 grid pointsfor comparison.

∂t A = B · ∇U + η∇⊥2 A,Fig. 2 Time evolution of the average flow and safety factor.

with ν = η = 1.2 × 10−4 for comparison, which shows
good convergence of the linear growth rate with respect todiﬀusivity and resolution. The linear instability grows exponentially until t  100. After this time, the system isfully nonlinear.
The time evolution of the average velocity and safety
factor q are shown in Fig. 2. It should be stressed thatthe initial reversal of the magnetic shear is essential forthe instability. Since JT-60U often experiences disruptions
especially when qmin crosses 2, we assumed initial qmin =1.7 and considered the (2, 1) mode. As is seen from thefigure, the instability tends to lower the q value just outsideof q ∼ 2 surfaces. At the same time, it raises the centralq(0). The instability doesn’t alter the profile of velocity
and safety factor in the outer region |x|  0.6, but it flattens
and makes them approach zero in the central region in thelinear regime.
When the instability comes into its nonlinear stage att  100, the enstrophy grows rapidly, and shows a somewhat complicated behavior. The average velocity and qprofile at t = 333 are flattened in a fairly wide range, andare close to zero in |x|  0.5. At t  380, when the enstrophy growth peaks, the q-profile is raised further, withq > 2 almost everywhere by t = 433.
Because of the global rearrangement of the current(2)with stream function U and flux function A introduced byu = ∇U × ez ,B = ∇A × ez + Bz ez ,(3)

where space is normalized by half system size L in the radial direction, time is normalized by L/vAy , and vAy denotes
the Alfvén velocity defined by the y (poloidal) componentof the magnetic field. The single-helicity calculation may
be justified because there is only a single set of unstable
low wave numbers in the example shown here. The inclusion of another dimension should not alter the qualitative
results. Our boundary condition corresponds to the ideal,no-slip wall at x = ±1, and poloidal periodicity.
Figure 1 shows the time evolution of various integral quantities. The straight line in the evolution of theL2 norm of perturbed stream function ||U1 || shows the exponential growth corresponding to the growth rate 0.045
obtained from an independent numerical eigenvalue analysis of the ideal (ν = η = 0) system, which will be presented elsewhere. We also showed an evolution of ||U1 ||015-2

Plasma and Fusion Research: Rapid Communications

Volume 1, 015 (2006)

profile, we believe that the instability is a good candidate
to explain some of the disruptive events observed in JT60U. The necessary flow is about 20-40% of the poloidal
Alfvén velocity, and the growth rate is one order of magnitude smaller than the flow shear rate in the example shownhere.
We acknowledge fruitful discussions with Drs. Yasutomo Ishii and Nobuyuki Aiba. This research is in part
supported by US DOE Center for Multiscale Plasma Dynamics.

[1] S. Takeji et al., Nucl. Fusion 42, 5 (2002).
[2] M.E. Stern, Phys. Fluids 6, 636 (1959).
[3] J.W.S. Rayleigh, Proc. London Math. Soc. 9, 57 (1880).
[4] A. Kent, Phys. Fluids 9, 1286 (1966); J. Plasma Phys. 2, 543(1968).
[5] X.L. Chen and P.J. Morrison, Phys. Fluids B 3, 863 (1991).
[6] F. Volponi, Z. Yoshida and T. Tatsuno, Phys. Plasmas 7, 2314
(2000); T. Tatsuno, F. Volponi and Z. Yoshida, Phys. Plasmas
8, 399 (2001).
[7] A. Ito et al., Phys. Plasmas 9, 4856 (2002).
[8] H.R. Strauss, Phys. Fluids 19, 134 (1976).

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